Internal problem ID [745]
Book: Elementary differential equations and boundary value problems, 10th ed., Boyce and
DiPrima
Section: Chapter 5.3, Series Solutions Near an Ordinary Point, Part II. page 269
Problem number: 10.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Gegenbauer, [_2nd_order, _linear, _with_symmetry_[0,F(x)]]]
Solve \begin {gather*} \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime } x +y \alpha ^{2}=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.005 (sec). Leaf size: 71
Order:=6; dsolve((1-x^2)*diff(y(x),x$2)-x*diff(y(x),x)+alpha^2*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1-\frac {\alpha ^{2} x^{2}}{2}+\frac {\alpha ^{2} \left (\alpha ^{2}-4\right ) x^{4}}{24}\right ) y \relax (0)+\left (x -\frac {\left (\alpha ^{2}-1\right ) x^{3}}{6}+\frac {\left (\alpha ^{4}-10 \alpha ^{2}+9\right ) x^{5}}{120}\right ) D\relax (y )\relax (0)+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 88
AsymptoticDSolveValue[(1-x^2)*y''[x]-x*y'[x]+\[Alpha]^2*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {\alpha ^4 x^5}{120}-\frac {\alpha ^2 x^5}{12}+\frac {3 x^5}{40}-\frac {\alpha ^2 x^3}{6}+\frac {x^3}{6}+x\right )+c_1 \left (\frac {\alpha ^4 x^4}{24}-\frac {\alpha ^2 x^4}{6}-\frac {\alpha ^2 x^2}{2}+1\right ) \]